Summary: A Class of Bases in L2
for the
Sparse Representation of Integral Operators
Bradley K. Alpert
3
Abstract
A class of multi-wavelet bases for L2
is constructed with the property
that a variety of integral operators is represented in these bases as sparse
matrices, to high precision. In particular, an integral operator K whose
kernel is smooth except along a nite number of singular bands has a sparse
representation. In addition, the inverse operator (I0K)01
appearing in the
solution of a second-kind integral equation involving K is often sparse in
the new bases. The result is an order O(n log2
n) algorithm for numerical
solution of a large class of second-kind integral equations.
Key Words. wavelets, integral equations, sparse matrices
AMS(MOS) subject classications. 42C15, 45L10, 65R10, 65R20
Families of functions ha;b,